Finite Group Databasebeta

← #27273128

Subgroups — #27273128

Order 13568.

418 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20276273C211yesmaximal, characteristic, complemented
2#20276186C211yesmaximal, characteristic, complemented
3#20276118C211yesmaximal, characteristic
4#20272538C211yesmaximal, characteristic, complemented
5#20272538C211yesmaximal, characteristic, complemented
6#20272536C211yesmaximal, characteristic
7C53×#2724C211yesmaximal, characteristic
8C2×#13133150—21no—
9C2×#13133138—21no—
10#16810493C2211yescharacteristic
11#16810385—21no—
12#16810385—21no—
13#16810385—21no—
14#16810385—21no—
15#16810378C2211yescharacteristic
16#16810378C2211yescharacteristic
17#16810378C2211yescharacteristic
18C2×#123×C53—21no—
19C53×#398—21no—
20C53×#397—21no—
21C53×#397—21no—
22C53×#390C2211yescharacteristic
23C53×#389C2211yescharacteristic
24C53×#389C2211yescharacteristic
25C53×#385—21no—
26C53×#385—21no—
27C22×#88322—21no—
28C22×#88322—21no—
29C22×#88322—21no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53—21no—
33#13133150—42no—
34#13133138—42no—
35C22×D4×C53D411yescharacteristic
36C23×C4×C53—21no—
37#123×C53—42no—
38#123×C53—42no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53D411yescharacteristic
43C2×C4⋊C4×C53D411yescharacteristic
44C2×C4⋊C4×C53D411yescharacteristic
45C2×C4⋊C4×C53D411yescharacteristic
46C2×C4⋊C4×C53D411yescharacteristic
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53—21no—
52C2×C42×C53C2311yescharacteristic
53#95×C53—21no—
54#95×C53—21no—
55#95×C53—21no—
56#95×C53—21no—
57#95×C53—21no—
58C24×C53—21no—
59C2×D4×C53—21no—
60C2×D4×C53—41no—
61C2×D4×C53—41no—
62C2×D4×C53D812yes—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—42no—
69C22×C4×C53—81no—
70C22×C4×C53—81no—
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×Dic53—41no—
75C4⋊C4×C53—21no—
76C4⋊C4×C53—21no—
77C4⋊C4×C53—21no—
78C4⋊C4×C53—21no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53—41no—
86C4⋊C4×C53—81no—
87C22⋊C4×C53—42no—
88C22⋊C4×C53—42no—
89C22⋊C4×C53—42no—
90C22⋊C4×C53—41no—
91C22⋊C4×C53—41no—
92C22⋊C4×C53—41no—
93C22⋊C4×C53—41no—
94C42×C53—41no—
95C4⋊Dic53—42no—
96C4⋊Dic53—41no—
97C4⋊Dic53—41no—
98C4⋊Dic53—41no—
99C4⋊Dic53—41no—
100C2×#88322—22no—
101C2×#88322—41no—
102C2×#88322—41no—
103C2×#88322—41no—
104C2×#88322—41no—
105C2×#88322—41no—
106C2×#88322—41no—
107C2×#88322—41no—
108C2×#88322—41no—
109C23×C53—22no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53—41no—
116C23×C53#12011yescharacteristic
117D4×C53—42no—
118D4×C53—81no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—84no—
141C2×C4×C53C2×D811yescharacteristic
142C2×C4×C53#13511yescharacteristic
143C2×C4×C53#13511yescharacteristic
144C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C53#13711yescharacteristic
147C2×Dic53—81no—
148C2×Dic53—81no—
149C2×Dic53—81no—
150#88322—42no—
151#88322—81no—
152#88322—81no—
153#7788—531nomaximal, Sylow
154C22×C53—42no—
155C22×C53—42no—
156C22×C53—42no—
157C22×C53—81no—
158C22×C53—81no—
159C22×C53—81no—
160C22×C53—81no—
161C22×C53#44711yescharacteristic
162C22×C53#44911yescharacteristic
163C22×C53#45211yescharacteristic
164C22×C53#45411yescharacteristic
165C22×C53#45511yescharacteristic
166C22×C53#45611yescharacteristic
167C22×C53#45811yescharacteristic
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—21no—
171C4×C53—21no—
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—81no—
175C4×C53—81no—
176C4×C53—81no—
177C4×C53—81no—
178C4×C53—81no—
179Dic53—161no—
180#2724D5311yescharacteristic
181#2099—531no—
182#2083—531no—
183#2071—531no—
184#1271—531no—
185#1271—531no—
186#1270—531no—
187C2×C53—82no—
188C2×C53#160711yescharacteristic
189C2×C53#160911yescharacteristic
190C2×C53#161511yescharacteristic
191C2×C53#162511yescharacteristic
192C2×C53#164011yescharacteristic
193C2×C53#164711yescharacteristic
194C2×C53#166411yescharacteristic
195C2×#123—21no—
196C2×#107—1061no—
197C2×#102—1061no—
198#398—21no—
199#397—21no—
200#397—21no—
201#390C2×D5311yescharacteristic
202#389C2×D5311yescharacteristic
203#389C2×D5311yescharacteristic
204#385—21no—
205#385—21no—
206#384—531no—
207#340—1061no—
208#340—1061no—
209#340—1061no—
210#340—1061no—
211#336—531no—
212#336—531no—
213#336—531no—
214C53#778811yescharacteristic, complemented, Sylow
215C22×D4C22⋊D5311yescharacteristic
216C23×C4—21no—
217C22×C8—1061no—
218C22×C8—1061no—
219C22×C8—1061no—
220#123—42no—
221#123—42no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—1061no—
226C2×C4⋊C4—1061no—
227C2×C4⋊C4—1061no—
228C2×C4⋊C4C22⋊D5311yescharacteristic
229C2×C4⋊C4C22⋊D5311yescharacteristic
230C2×C4⋊C4C22⋊D5311yescharacteristic
231C2×C4⋊C4C22⋊D5311yescharacteristic
232C2×C4⋊C4C22⋊D5311yescharacteristic
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C42C22×D5311yescharacteristic
239#107—2122no—
240#102—2122no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—21no—
246C24—21no—
247C2×D4—21no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4#1171992212yes—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—42no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4—2121no—
260C22×C4C2×C22⋊D5311yescharacteristic
261C22×C4C2×C22⋊D5311yescharacteristic
262C22×C4C2×C22⋊D5311yescharacteristic
263C2×C8—1062no—
264C2×C8—2121no—
265C2×C8—2121no—
266C2×C8—2121no—
267C2×C8—2121no—
268C2×C8—2121no—
269C2×C8—2121no—
270C2×C8—2121no—
271C2×C8—2121no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—81no—
284C4⋊C4—2122no—
285C4⋊C4—2121no—
286C4⋊C4—2121no—
287C4⋊C4—2121no—
288C4⋊C4—2121no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C42—41no—
297C23—22no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23#1313318611yescharacteristic
305D4—42no—
306D4—81no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—84no—
329C2×C4—4241no—
330C2×C4—4241no—
331C2×C4—4241no—
332C2×C4C2×#1171992211yescharacteristic
333C2×C4#1313323711yescharacteristic
334C2×C4#1313323711yescharacteristic
335C2×C4#1313324211yescharacteristic
336C2×C4#1313324311yescharacteristic
337C2×C4#1313324711yescharacteristic
338C8—4241no—
339C8—4241no—
340C8—2122no—
341C22—42no—
342C22—42no—
343C22—42no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22#1681075711yescharacteristic
349C22#1681076811yescharacteristic
350C22#1681078511yescharacteristic
351C22#1681079611yescharacteristic
352C22#1681080111yescharacteristic
353C22#1681080611yescharacteristic
354C22#1681081311yescharacteristic
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—81no—
362C4—81no—
363C4—81no—
364C4—81no—
365C4—81no—
366C4—8481no—
367C2—82no—
368C2#2027370111yescharacteristic
369C2#2027371211yescharacteristic
370C2#2027374611yescharacteristic
371C2#2027380411yescharacteristic
372C2#2027390111yescharacteristic
373C2#2027395011yescharacteristic
374C2#2027404911yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2724C2
socleC23×C53#120
prime radicalC22×D4×C53D4
hypercenter#2724D53
hypocenterC53#7788
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2724

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #20276273C2semidirect
2. #20276186C2semidirect
4. #20272538C2semidirect
5. #20272538C2semidirect
214. C53#7788semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202762732
#202761862
#202761182
#202725382
#202725362
C53×#27242
#778853